Choice A is the correct answer.

1. Square both sides: $2x + 3 = x^2$.
2. Rearrange: $x^2 - 2x - 3 = 0$.
3. Factor: $(x - 3)(x + 1) = 0$.
4. Potential roots: $3, -1$.
5. Check: $\sqrt{2(3)+3} = \sqrt{9} = 3$ (Matches x). $\sqrt{2(-1)+3} = \sqrt{1} = 1 eq -1$. So -1 is extraneous.

Choice B is incorrect because -1 is extraneous. Choice C is incorrect. Choice D is incorrect.